Given Problem
Yusuf_Uysal2 shared a problem where for positive real numbers (x) and (y), the following equations are given:
- (3x^2 - 2y^2 = 12)
- (x^2 + 4y^2 = 18)
You are asked to find the ratio (\frac{x}{y}).
Answer:
To find (\frac{x}{y}), let’s manipulate the given equations. We will solve them simultaneously to find the values of (x) and (y), and their subsequent ratio.
Step 1: Solve the Equations
Equation 1:
[ 3x^2 - 2y^2 = 12 ]
Equation 2:
[ x^2 + 4y^2 = 18 ]
To solve these equations, let’s eliminate one of the variables, usually best accomplished by methodically manipulating and adding or subtracting the equations.
Multiply Equation 2 by 2:
[ 2(x^2 + 4y^2) = 2 \times 18 ]
[ 2x^2 + 8y^2 = 36 ] … (Equation 3)
Subtract Equation 1 from Equation 3:
[
(2x^2 + 8y^2) - (3x^2 - 2y^2) = 36 - 12
]
[
2x^2 + 8y^2 - 3x^2 + 2y^2 = 24
]
[
-x^2 + 10y^2 = 24
]
[
-x^2 + 10y^2 = 24 \quad \Rightarrow \quad x^2 = 10y^2 - 24
]
Now, substitute ( x^2 = 10y^2 - 24 ) into Equation 2:
[
(10y^2 - 24) + 4y^2 = 18
]
[
14y^2 - 24 = 18
]
[
14y^2 = 42
]
[
y^2 = 3
]
[
y = \sqrt{3}
]
Step 2: Substitute back to find (x)
Substitute (y = \sqrt{3}) back into the expression for (x^2):
[
x^2 = 10(\sqrt{3})^2 - 24
]
[
x^2 = 30 - 24
]
[
x^2 = 6
]
[
x = \sqrt{6}
]
Step 3: Calculate (\frac{x}{y})
Now, compute (\frac{x}{y}):
[
\frac{x}{y} = \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{\frac{6}{3}} = \sqrt{2}
]
Final Answer:
The ratio (\frac{x}{y}) is (\sqrt{2}), which corresponds to option B.